Binomial Distribution Calculator
Calculate probabilities for a series of independent trials.
Probability Results
P(X = k)
0.000
P(X ≤ k)
0.000
P(X ≥ k)
0.000
About the Binomial Distribution Calculator
This calculator helps you determine the probability of a specific number of successful outcomes in a fixed set of trials. It's a fundamental tool in statistics and probability, useful in fields ranging from quality control in manufacturing to predicting outcomes in sports. Simply enter your parameters to see the likelihood of various scenarios.
Formula Explained
The calculator uses the binomial probability formula to find the probability of exactly 'k' successes in 'n' trials:
- C(n, k) is the number of combinations (n choose k).
- p is the probability of success on a single trial.
- n is the total number of trials.
- k is the number of successes.
How to Interpret the Results
Understanding your results can provide powerful insights:
Analyze the "Peak"
The highest bar on the chart represents the most likely number of successes. This is the expected outcome of your experiment.
Assess the Spread
A wide, flat distribution means outcomes are highly variable. A narrow, steep distribution indicates results will be very consistent and close to the average.
Frequently Asked Questions
What is a binomial distribution? →
A binomial distribution is a specific type of probability distribution used in statistics. It describes the number of successful outcomes in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure. For example, it can be used to model the number of heads you get if you flip a coin 10 times, or the number of defective products in a batch of 50.
What are the parameters of a binomial distribution? →
A binomial distribution is defined by two key parameters: 'n', which is the total number of trials, and 'p', which is the probability of success on any single trial. The probability of failure is simply 1 - p. For a coin flip, n might be 10 (10 flips) and p would be 0.5 (a 50% chance of heads).
What is the difference between P(X=k) and P(X≤k)? →
P(X=k) represents the probability of getting *exactly* 'k' successes. For example, the chance of getting exactly 7 heads in 10 coin flips. P(X≤k) is the *cumulative* probability of getting 'k' or *fewer* successes. This means it's the sum of the probabilities of getting 0 successes, 1 success, 2 successes, all the way up to k successes. Our calculator provides both values for a complete analysis.
When would you use a binomial distribution calculator? →
You would use it in any scenario that meets four conditions: a fixed number of trials, each trial is independent, there are only two outcomes, and the probability of success is constant. Common examples include quality control (number of defective items in a batch), medical trials (number of patients responding to a treatment), or sports (number of free throws made out of a certain number of attempts).
What are the mean and variance of a binomial distribution? →
The mean (or expected value) of a binomial distribution tells you the average number of successes you can expect. It is calculated with the simple formula: μ = n * p. The variance measures how spread out the results are likely to be. It is calculated as: σ² = n * p * (1 - p).